Question 1

For the rational function $$f(x) = \frac{x-4}{x^2$ + 2x - 8}$$, what is the equation of the vertical asymptote(s)?

Accepted Answer

$$x = 2$$ and $$x = -4$$

Question 2

Given $$f(x) = \frac{x-4}{x^2$ + 2x - 8}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 3

Differentiate $$f(x) = \frac{1}{3} x^3$ \ln x$$ with respect to $$x.$$

Accepted Answer

$$x^2$$ \ln x + $$x^2$$

Question 4

A spherical balloon is inflating with helium at a rate of $$1.4\ 	ext{liters/min}. $$How fast is the balloon's radius increasing at the instant the radius is $$3\ 	ext{m}$$? (Volume of sphere: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{1.4}{4\pi (3^2$)}$$

Question 5

Implicitly differentiate $$y^2$$ - $$x^2 = 28$ and find $$\frac{dy}{dx}$$ at the point (2$$, -6)$$. What is the value?

Accepted Answer

$$\frac{2}{-6}$$

Question 6

A restaurant purchased kitchen equipment on January 1, 2017. On January 1, 2018, the value was $$14,600. The value after that date is modeled as V(t) = 14$$,$$600e^{-0.1t}$. What is the rate of change in value on January 1, 2019?

Accepted Answer

$$-1,460e$$^{-0.2}$$

Question 7

For $$f(x) = 2x^3$ - 6x^2$ + 1$$, what are the $$x-$$coordinates of the critical points?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 8

For $$f(x) = 2x^3$ - 6x^2$ + 1$$, on which interval is the function increasing?

Accepted Answer

$$x \u003e 2$$

Question 9

Use the Chain Rule to differentiate $$y = (2x + 3)^5$.

Accepted Answer

$$10(2x+3)^4$$

Question 10

Implicitly differentiate $$x^y + y = \ln y to $$find $$\frac{dy}{dx}.$$

Accepted Answer

$$\frac{-y}{x^y \ln x + 1 - \frac{1}{y}}$$

Question 11

Find the derivative of $$g(x) = \frac{7x-8}{4x^2$}.$$

Accepted Answer

$$\frac{7}{4x^2$} - \frac{7x-8}{2x^3$}$$

Question 12

Find the derivative of $$f(x) = \sqrt{\frac{7x}{x}}.$$

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 13

For $$f(x) = \frac{x-4}{x^2$ - 5x - 6}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq -2, 3$$

Question 14

For $$f(x) = \frac{x-4}{x^2$ - 5x - 6}$$, what are the vertical asymptotes?

Accepted Answer

$$x = -2$$ and $$x = 3$$

Question 15

For $$f(x) = \frac{x-4}{x^2$ - 5x - 6}$$, what is the horizontal asymptote?

Accepted Answer

$$y = 0$$

Question 16

For $$f(x) = \frac{x-4}{x^2$ - 5x - 6}$$, what is the $$y-$$intercept?

Accepted Answer

$$(0, \frac{-4}{-6})$$

Calculus Final Exam Study Guide: Key Concepts and Applications

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